A really intriguing answer is brought in the Aruch Hasulchan (417:6-9) That it is based on a Yerushalmi in Sanhedrin 5:3 Where it brings a case where 2 people came to testify about a crime and one says it was the Second day of the Month another says it was the third and we cant disprove their testimony because maybe one knew that the Month had an extra day and the second did not, there Reb Yosa says "like me when once I did not say Mussaf. I did not know when Rosh Chodesh was".This could be construed to mean he missed it because he did not Daven Mussaf the Shabbos before in shul so that's why he missed Rosh Chodesh Because he missed Rosh Chodesh bentching and Hence Mussaf thereafter a semi proof for our Minhag of saying it a shabbos before.
More of what was said earlier cant be found in the Levush in Siman 481 says "The custom is to bless every new month on the Shabbos before Rosh Chodesh so the Matter is Publicized and everyone knows on what day Rosh chodesh will be". That is since there are more people in shul on Shabbos it is the optimal time.